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arXiv:2108.05922 · Physical Review X

Anomalous Dimensions of Monopole Operators at the Transitions between Dirac and Topological Spin Liquids

Éric Dupuis, Rufus Boyack, William Witczak‐Krempa

Published Jul 19, 2022• 20 citations
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Abstract

Monopole operators are studied in a large family of quantum critical points between Dirac spin liquids and topological quantum spin liquids (QSLs): chiral and Z 2 QSLs.These quantum phase transitions are described by conformal field theories (CFTs): quantum electrodynamics in 2 þ 1 dimensions with 2N flavors of two-component massless Dirac fermions and a four-fermion interaction term.For the transition to a chiral spin liquid, it is the Gross-Neveu interaction (QED 3 -GN), while for the transitions to Z 2 QSLs, it is a superconducting pairing term with general spin and valley structure (generalized QED 3 -Z 2 GN).Using the state-operator correspondence, we obtain monopole scaling dimensions to subleading order in 1=N.For monopoles with a minimal topological charge q ¼ 1=2, the scaling dimension is 2N × 0.26510 at leading order, with the quantum correction being 0.118911(7) for the chiral spin liquid, and 0.102846(9) for the simplest Z 2 case (the expression is also given for a general pairing term).Although these two anomalous dimensions are nearly equal, the underlying quantum fluctuations possess distinct origins.The analogous result in QED 3 is also obtained, and we find a subleading contribution of -0.038138ð5Þ, which is slightly different from the value -0.0383 first obtained in the literature.The scaling dimension of a QED 3 -GN monopole with minimal charge is very close to the scaling dimensions of other operators predicted to be equal by a conjectured duality between QED 3 -GN with 2N ¼ 2 flavors and the CP 1 model.Additionally, nonminimally charged monopoles with equal charges on both sides of the duality have similar scaling dimensions.By studying the large-q asymptotics of the scaling dimensions in QED 3 , QED 3 -GN, and QED 3 -Z 2 GN, we verify that the constant Oðq 0 Þ coefficient precisely matches the universal nonperturbative prediction for CFTs with a global U(1) symmetry.Finally, we identify numerous open questions regarding the fate of monopoles and their hierarchies at transitions to spin liquids and ordered phases.

DOI
10.1103/physrevx.12.031012

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Monopole operators are studied in a large family of quantum critical points between Dirac spin liquids and topological quantum spin liquids (QSLs): chiral and Z 2 QSLs.These quantum phase transitions are described by conformal field theories (CFTs): quantum electrodynamics in 2 þ 1 dimensions with 2N flavors of two-component massless Dirac fermions and a four-fermion interaction term.For the transition to a chiral spin liquid, it is the Gross-Neveu interaction (QED 3 -GN), while for the transitions to Z 2 QSLs, it is a superconducting pairing term with general spin and valley structure (generalized QED 3 -Z 2 GN).Using the state-operator correspondence, we obtain monopole scaling dimensions to subleading order in 1=N.For monopoles with a minimal topological charge q ¼ 1=2, the scaling dimension is 2N × 0.26510 at leading order, with the quantum correction being 0.118911(7) for the chiral spin liquid, and 0.102846(9) for the simplest Z 2 case (the expression is also given for a general pairing term).Although these two anomalous dimensions are nearly equal, the underlying quantum fluctuations possess distinct origins.The analogous result in QED 3 is also obtained, and we find a subleading contribution of -0.038138ð5Þ, which is slightly different from the value -0.0383 first obtained in the literature.The scaling dimension of a QED 3 -GN monopole with minimal charge is very close to the scaling dimensions of other operators predicted to be equal by a conjectured duality between QED 3 -GN with 2N ¼ 2 flavors and the CP 1 model.Additionally, nonminimally charged monopoles with equal charges on both sides of the duality have similar scaling dimensions.By studying the large-q asymptotics of the scaling dimensions in QED 3 , QED 3 -GN, and QED 3 -Z 2 GN, we verify that the constant Oðq 0 Þ coefficient precisely matches the universal nonperturbative prediction for CFTs with a global U(1) symmetry.Finally, we identify numerous open questions regarding the fate of monopoles and their hierarchies at transitions to spin liquids and ordered phases.
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We grow the web of dualities for conformal field theories (CFTs) in 2+1 spacetime dimensions to include quantum critical transitions of Dirac fermions. Our construction uses the seed duality, equating a free Dirac fermion with a complex boson coupled to a Chern-Simons gauge field, to express various Gross-Neveu-Yukawa (GNY) critical points of N fermions in terms of the same number of complex bosons and partner gauge fields. In some instances, monopole operators appear explicitly in the dual action, such as for the superconducting transition of a single Dirac fermion, which is a CFT with emergent supersymmetry. We match phase diagrams, symmetries and their anomalies. By further gauging certain symmetries, we derive offspring dualities. For example, starting with the duality for the regular GNY transition of 2 Dirac fermions, we arrive at the duality between a Neel-to-VBS deconfined quantum critical point and a gauged-Gross-Neveu transition. Finally, we use a large-N expansion for monopole operators, and compare with known scaling dimensions in the simplest single-flavour GNY transition.
The fermion disorder operator has been shown to reveal the entanglement information in 1D Luttinger liquids and 2D free and interacting Fermi and non-Fermi liquids emerging at quantum critical points (QCPs) [W. Jiang et al., arXiv:2209.07103]. Here we study, by means of large-scale quantum Monte Carlo simulation, the scaling behavior of the disorder operator in correlated Dirac systems. We first demonstrate the logarithmic scaling behavior of the disorder operator at the Gross-Neveu (GN) chiral Ising and Heisenberg QCPs, where consistent conformal field theory (CFT) content of the GN-QCP in its coefficient is found. Then we study a 2D monopole-free deconfined quantum critical point (DQCP) realized between a quantum-spin Hall insulator and a superconductor. Our data point to negative values of the logarithmic coefficients such that the DQCP does not correspond to a unitary CFT. Density matrix renormalization group calculations of the disorder operator on a 1D DQCP model also detect emergent continuous symmetries.
We give evidence for the web of 3d bosonization dualities in conformal field theories (CFTs) by computing monopole operator scaling dimensions in (2+12+12+1)-dimensional quantum electrodynamics (QED3) with Chern-Simons level kkk and NNN complex bosons in a large-NNN, kkk expansion. We first consider the k=0k=0k=0 case, where we show that scaling dimensions previously computed to subleading order in 1/N1/N1/N can be extrapolated to N=1N=1N=1 and matched to O(2)O(2)O(2) Wilson-Fisher CFT scaling dimensions with around 5% error, which is evidence for particle-vortex duality. We then generalize the subleading calculation to large NNN, kkk and fixed k/Nk/Nk/N, extrapolate to N=k=1N=k=1N=k=1, and consider monopole operators that are conjectured to be dual to nondegenerate scalar operators in a theory of a single Dirac fermion. We find matches typically with 1% error or less, which is strong evidence of this so-called ``seed'' duality that implies a web of 3d bosonization dualities among CFTs.
The Dirac spin liquid (DSL) is a two-dimensional (2D) fractionalized Mott insulator featuring massless Dirac spinon excitations coupled to a compact U(1) gauge field, which allows for flux-tunneling instanton events described by magnetic monopoles in (2+1)D Euclidean spacetime. The state-operator correspondence of conformal field theory has been used recently to define associated monopole operators and determine their quantum numbers, which encode the microscopic symmetries of conventional ordered phases proximate to the DSL. In this work, we utilize semiclassical instanton methods not relying on conformal invariance to construct monopole operators directly in (2+1)D spacetime as instanton-induced ’t Hooft vertices, i.e., fermion-number-violating effective interactions originating from zero modes of the Euclidean Dirac operator in an instanton background. In the presence of a flavor-adjoint fermion mass, resummation of the instanton gas is shown to select the correct monopole to be proliferated, in accordance with predictions of the state-operator correspondence. We also show that our instanton-based approach is able to determine monopole quantum numbers on bipartite lattices.
We study the universal critical properties of the QED3{\mathrm{QED}}_{3}QED3​-Gross-Neveu-Yukawa model with NNN flavors of four-component Dirac fermions coupled to a real scalar order parameter at four-loop order in the $\ensuremath{\epsilon}$ expansion below four dimensions. For N=1N=1N=1, the model is conjectured to be infrared dual to the SU(2)\text{SU}(2)SU(2)-symmetric noncompact CP1\mathbb{C}{\mathrm{P}}^{1}CP1 model, which describes the deconfined quantum critical point of the N\'eel-valence-bond-solid transition of spin-1/2 quantum antiferromagnets on the two-dimensional square lattice. For N=2N=2N=2, the model describes a quantum phase transition between an algebraic spin liquid and a chiral spin liquid in the spin-1/2 kagome antiferromagnet. For general NNN we determine the order-parameter anomalous dimension, the correlation length exponent, the stability critical exponent, as well as the scaling dimensions of SU(N)\text{SU}(N)SU(N) singlet and adjoint fermion mass bilinears at the critical point. We use Pad\'e approximants to obtain estimates of critical properties in 2+1 dimensions.